UNCOUNTABLE SATURATED STRUCTURES HAVE THE SMALL INDEX PROPERTY

We prove the following theorem. Let M be a, uncountable saturated stru cture of cardinality lambda = lambda<lambda and assume that G is a sub group of Aut (M) whose index is less than or equal to lambda. Then the re exists a subset A of cardinality strictly less than lambda such tha t every automorphism of M leaving A pointwise fixed is in G.